The singular "time" here means one occurrence or event or "occasion" as Oxford mentions. "At a time" means "during each event." There may be some ambiguity in certain cases as others have pointed out, but the main idea can generally be determined from the context. If I take my pills four at a time, that could mean I'm swallowing all four pills in one gulp or it could mean I'm swallowing one pill and then immediately swallowing the next pill after it and so on but the main idea is that I'm not spreading the pills out through the day (meaning I'm not taking two in the morning and two at night for example).
We had to go and see the principal one at a time. In this case one "time" is one session with the principal. No two students were seeing the principal simultaneously. Each talk with the principal involved only one student.
He surfs the internet for hours at a time. Similarly, a "time" here is an Internet session. Rather than using the Internet for just a few minutes before going to do something else, he spends consecutive hours on the Internet between each break.
She ran up the stairs two at a time. Like others have said, each "time" is each stride/step she takes. So during each stride (each occurrence of either foot landing) she will ascend two stair steps instead of one. This isn't ambiguous in terms of how many people are involved because "she" is singular, but the terms we use like "stair" and "step" can be ambiguous. "Stair" can mean a whole flight of stairs but we know it means a single step in this case and I've said "stair step" to be extra clear. "Step" is also ambiguous because it can mean a single stair step in a staircase or it can refer to the action of stepping as in "take a step forward" but I've tried to be clear in my meaning.